cp's OEIS Frontend

A012816 E.g.f. arctan(sec(x)*sinh(x)) (odd powers only).

%I A012816 #38 Jul 25 2024 18:02:27
%S A012816 1,2,-20,-488,22160,1616672,-172976960,-25518205568,4964227109120,
%T A012816 1231298393825792,-379260096755225600,-142026494757146421248,
%U A012816 63547531933929827962880,33481297996129270926221312,-20517021964757071715832381440,-14468510293983989090015078678528
%N A012816 E.g.f. arctan(sec(x)*sinh(x)) (odd powers only).
%C A012816 The unsigned sequence {|a(n)|}n>=1 = [2,20,488,22160,...] enumerates binary increasing trees on 2*n vertices with a perfect matching (Kuba and Wagner).
%H A012816 Alois P. Heinz, <a href="/A012816/b012816.txt">Table of n, a(n) for n = 0..228</a>
%H A012816 M. Kuba and S. Wagner, <a href="http://www.emis.de/journals/SLC/wpapers/s57kubawagn.html">Perfect matchings and k-indecomposabilty of increasing trees</a>, Seminaire Lotharingien de Combinatoire 57 (2007), Article B57a.
%F A012816 1/cosh(x*sqrt(2)) = 1 - 2x^2/2! + 20*x^4/4! - 488*x^6/6! +-...
%F A012816 a(n) = (-1)^[n/2]*2^n*A000364(n). - _Philippe Deléham_, Jun 16 2007
%F A012816 G.f. (for the unsigned sequence): 1/G(0) where G(k) =  1 - 2*x*(k+1)^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 12 2013
%F A012816 G.f. (for the unsigned sequence): Q(0), where Q(k) = 1 - 2*x*(k+1)^2/(2*x*(k+1)^2 - 1/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 10 2013
%F A012816 E.g.f.(for the unsigned sequence, odd powers only): 1 + T(0)*x^2 /(1-x^2), where T(k) = 1 - 2*x^2*(2*k+1)*(2*k+2)/( 2*x^2*(2*k+1)*(2*k+2) + ((2*k+1)*(2*k+2)-2*x^2)*((2*k+3)*(2*k+4)-2*x^2)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 14 2013
%e A012816 arctan(sec(x)*sinh(x)) = x+2/3!*x^3-20/5!*x^5-488/7!*x^7+22160/9!*x^9...
%p A012816 a:= n-> (2*n+1)! *
%p A012816     coeff(series(arctan(sec(x)*sinh(x)), x, 2*(n+1)), x, 2*n+1):
%p A012816 seq(a(n), n=0..20);
%t A012816 With[{nn=40},Take[CoefficientList[Series[ArcTan[Sec[x]Sinh[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Jul 25 2024 *)
%Y A012816 Bisection (odd part) of A009342.
%K A012816 sign
%O A012816 0,2
%A A012816 Patrick Demichel (patrick.demichel(AT)hp.com)